Probability > Disjoint Events
What are Disjoint Events?
Disjoint events cannot happen at the same time. In other words, they are mutually exclusive. Disjoint events are disjointed, or not connected. Another way of looking at disjoint events are that they have no outcomes in common. This definition can be easier to wrap your head around if you think of the opposite: overlapping events have one or more outcomes in common. Put in formal terms, events A and B are disjoint if their intersection is zero:
P(A∩B) = 0.
You’ll sometimes see this written as:
P(A and B) = 0.
The two terms are equivalent. All disjoint events are dependent events [1]. This is because if we know event A and B cannot happen together, this implies that if one event happens, it affects the probability of the other event happening. For example, let’s say event A is winning the lottery and event B is buying a losing card. Picking a winning card might have a probability of 1 in 1,000, but as soon as you buy the losing card, your probability of winning is zero. Thus, the occurrence of one event has affected the other.
Examples of Disjoint Events
- A football game can’t be held at the same time as a rugby game on the same field.
- Heading East and West at the same time is impossible.
- Tossing a coin and getting a heads and a tails at the same time is impossible.
- You can’t take the bus and the car to work at the same time.
- You can’t get a pay raise and a pay decrease at the same time.
- Your dog can’t be both a purebred dog and a mutt.
The following Venn diagram shows two possible events (disjoint and overlapping) for rolling a single die. The diagram on the left shows that the intersection is zero (therefore, these are disjoint). As the diagram on the right does have an intersection, these events are not disjoint. The intersection of the two events is the empty set. 
Calculating probabilities for disjoint events
If two events A and B are disjoint, then the probability of either event is the sum of the probabilities of the two events [2]: P(A or B) = P(A) + P(B).
In other words, to find probabilities of disjoint events, add their probabilities together. For example, the probability of rolling a 1 on a six-sided die is 1/6 and the probability of rolling a 2, 3, 4, 5, or 6 is 5/6, then
P(1 or 2, 3, 4, 5, 6) = 1/6 + 5/6 = 1.
This is called the Addition Rule for Disjoint Events.
If two events A and B are not disjoint, then the probability of the event that either A or B happens (called their union) equals the sum of their probabilities minus the sum of their intersection [2].
For example, suppose the probability of event A, drawing a black ball from an urn on the first draw, is 3/5. Event B, drawing a white ball on the second draw, has a probability of 2/5. The probability of their intersection is 3/5 * 2/5 = 6/25. Thus, the probability of their union is 3/5 + 2/5 – 6/25 = 1 – 6/25 = 19/25 = 0.76.
If one or another disjoint event must happen, then the events are complementary events.
Mutually exclusive vs disjoint
Although disjoint events are sometimes called mutually exclusive events, there is a subtle difference between the terms disjoint and mutually exclusive: Disjoint is a property of sets while mutually exclusive is a property of events (sets in a probability space):
- Disjoint sets do not have any elements in common. For example, the set of odd numbers and the set of even numbers are disjoint sets.
- Mutually exclusive events are events that cannot happen at the same time. For example, the event of traveling in a car and flying in an airplane are mutually exclusive events.
If you’re studying probability and statistics, mutually exclusive and disjoint are, in practical terms, synonymous. However, if you’re working in set theory, you’ll need to understand the subtle differences between the two terms.
References
[1] UF Biostatistics. Conditional Probability and Independence [2] Easton, V. & McColl, J. Statistics Glossary.
[2] Easton, V. & McColl, J. Statistics Glossary.